Confidence intervals for maximal reliability in tau‐equivalent models

Abstract

Subjective probabilities play an important role in marketing research, for example where individuals rate the likelihood that they will purchase a new developed product. The tau‐equivalent model can describe the joint behaviour of multiple test items measuring the same subjective probability. In this paper we stress the use of confidence intervals to assess reliability, as this allows for a more critical assessment of the items as measurement instruments. To improve the reliability one can use a weighted sum as the outcome of the test rather than an unweighted sum. In principle, the weights may be chosen so as to obtain maximal reliability. We propose two new confidence intervals for the maximal reliability in the tau‐equivalent model and we compare these two new intervals with intervals derived earlier in Yuan and Bentler (Psychometrika, 67, 2002, 251) and Raykov and Penev (Multivariate Behavioral Research, 41, 2006, 15). The comparison involves coverage curves, a methodology that is new in the field of reliability. The existing Yuan–Bentler and Raykov–Penev intervals are shown to overestimate the maximal reliability, whereas one of our proposed intervals, the stable interval, performs very well. This stable interval hardly shows any bias, and has a coverage for the true value which is approximately equal to the confidence level.